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Does anyone know this result?

Let $A$ be a matrix symmetric matrix such that $\lambda I \le A(x) \le \Lambda I$ where $I$ is the identity and $v$ is a solution of $ \mbox{div}(A(x)\nabla v) = 0 $ in $B_R \subset \mathbb{R^{n}}$. Then for some constant $0<\alpha<1$ we have $$ \int_{B_r} | \nabla v - (\nabla v)_r|^{2}dx \le C(\lambda, \Lambda)\Bigl( \dfrac{r}{R}\Bigr)^{n - 2 + 2\alpha} \int_{B_r}| \nabla v(x) -(\nabla v)_{R}|^{2}dx $$ for any $0<r<R$.Where $(f)_{r}$ is the classical average notation \begin{equation} (f)_{r} = \dfrac{1}{|B_r|}\int_{B_r}f dx. \end{equation}

If you know this result or result similar, or you know some reference for this, please tell me.

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What is $(\nabla v)_r$? –  Dirk Jun 14 '12 at 17:40
    
I wrote above now. –  user29999 Jun 14 '12 at 17:57
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1 Answer

This kind of result is used to proof Hölder continuity of (the derivatives) of solutions to equations of the kind you wrote down. You'll find this kind of estimates in treatises on elliptic PDE. A classic but often difficult to read is C.B. Morrey's 'Multiple Integrals in the calculus of variations'. A book with the same title by Mariano Giuaquinta treats elliptic systems and contains many references. See in particular chapter II and III in there. One of the standard references is Gilbarg and Trudingers treatise 'Elliptic Partial Differential Equations of the Second Order', chapter 7 ff.

Online I found the this article, see corollary 2.2 for such a result, may be this gives you a place to start from. I did not check this article in detail, though.

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Thank you very much. –  user29999 Jun 14 '12 at 19:39
    
Search terms with which you may google additional results are 'morrey', 'estimate', 'divergence', 'hölder' and, obviously, combinations of these terms ;-) –  user20266 Jun 14 '12 at 19:45
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